Problem: Add the following rational expressions. $\dfrac{7m^4}{7m+2}+\dfrac{3m^2}{m+4}=$
Explanation: We can add two rational expressions whose denominators are equal by adding the numerators and keeping the denominator the same. [Does this fit with how we add rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({7m+2})\cdot({m+4})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{7m^4}{{7m+2}}+\dfrac{3m^2}{{m+4}} \\\\ &=\dfrac{7m^4\cdot({m+4})}{({7m+2})\cdot({m+4})}+\dfrac{3m^2\cdot({7m+2})}{({m+4})\cdot({7m+2})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's add! $\begin{aligned} &\phantom{=}\dfrac{7m^4\cdot(m+4)}{(7m+2)\cdot(m+4)}+\dfrac{3m^2\cdot(7m+2)}{(m+4)\cdot(7m+2)} \\\\ &=\dfrac{7m^4\cdot(m+4)+3m^2\cdot(7m+2)}{(7m+2)(m+4)} \\\\ &=\dfrac{7m^5+28m^4+21m^3+6m^2}{(7m+2)(m+4)} \end{aligned}$ In conclusion, $\dfrac{7m^4}{7m+2}+\dfrac{3m^2}{m+4}=\dfrac{7m^5+28m^4+21m^3+6m^2}{(7m+2)(m+4)}$